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1.2 2-Person, Zero-Sum Games

Two players Total payoff = 0, thus Payoff to Player I = – Payoff to Player II

Information can be summarised nicely in a simple payoff table.

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Player II

A1 A2 An

a1 v1,1 v1,2 v1,n

a2 v2,1 v2,2 v2,n

Player I ... ... ... ...

... ... ... ...

am vm,1 vm,n

Convention

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ai = ith action (pure strategy) for Player I Aj = jth action (pure strategy) for Player II

m = # of actions available to Player I n = # of actions available to Player II

vij = Payoff to Player I if they select ai and

Player II selects action Aj . Payoff to Player II = – Payoff to Player I

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Example

Player 1 has options: 2 pure strategies a1 or a2. Player 2 has options: 3 pure strategies A1, A2 or A3. If player 1 plays pure strategy a1 and Player 2 plays

pure strategy A3 then Player 1 wins 25 from Player 2.

A1 A2 A3

a12 -4 25

a26 10 -20

Player 1

Player 2

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Solution?

Player I Player II

Independent, unbiasedanalyst to determine what are the“best” strategies for the two players.

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1.2.1 Example

A1 A2 A3

a12 -4 25

a26 10 -20

What are the best strategies for the players?

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Conflict: Player 1 prefers (a1,A3) with payoff = 25 Player 2 prefers (a2,A3) with payoff = 20 But what would Player 2 do knowing

that Player 1 would go for the 25? What would Player 1 do ........?

What should the players do if they wish to “resolve” the game?

A1 A2 A3

a12 -4 25

a26 10 -20

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Player I Player II

I want the maximum payoff to Player I

I want the maximum payoff to Player II

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Player I Player II

I want the maximum payoff to Player I

I want the maximum payoff to Player II

Be sensible, guys!

Player I Player II

I want the maximum payoff to Player I

I want the maximum payoff to Player II

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Basic Issue

Sensible = ?

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Bottom Line

We need to agree on some fundamental principles to guide us in deciding what is best for the players.

There are two basic issues to be resolved: What kind of payoffs should the players expect? How do you guarantee that the players will

accept that solution?

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Payoff Ideal:

Maximum possible payoff to each player Minimum Expected: Each player gets what she can

“secure”, namely the maximum that she can get regardless of what the other player does.

We look at the worst things that can happen and pick the best of these.

Conservative approach. We seek a strategy such that changing

it could achieve a worse outcome.

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Philosophy

Conservative approach

We wish to be as secure as possible and not take the risk of larger losses in pursuit of larger gains.

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Old example

Ideal payoff :

Player 1 =25, Player 2 = 20 Maximum SECURED Payoff:

Player 1 = –4 ; Player 2 = –6 i.e. Player 1 can make sure of doing no worse than losing

4, (regardless of what Player 2 does) by playing a1.

A1 A2 A3

a12 -4 25

a26 10 -20

James Bond

Ziegfried

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1.2.2 Definition The quantity

si := min {vij: j=1,2,...,n} , i=1,2,...,m

is called the security level for Player I associated with strategy ai .

Similarly, the quantity

Sj := max {vij: i=1,2,...,m} , j=1,2,...,n

is called the security level for Player II associated with strategy Aj .

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Principle I

Player I acts to maximize their security level

Player II acts to minimize their security level

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1.2.3 Example

How do we compute the security levels?

A1 A2 A3

a1 1 10 3

a2 -2 5 1

a3 1 -8 1

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A1 A2 A3 si

a1 1 10 3 1

a2 -2 5 1 -2

a3 1 -8 1 -8

Sj 1 10 3

20

A1 A2 A3 si

a1 1 10 3 1

a2 -2 5 1 -2

a3 1 -8 1 -8

Sj 1 10 3

Will the players accept this solution? We apply the same idea to another example:

Maximum security level for Player I = 1. Minimum security level for Player II = 1. Suggested solution (a1, A1) .

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1.2.4 Example

The suggested solution (a2, A2) is not stable.

A1 A2 si

a1 1 5 1

a2 6 2 2

S j 6 5

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1.2.2 Definition (Page 10)

A solution (ai, Aj) to a zero-sum two-person game is stable (or in equilibrium) if Player I expecting Player II to Play Aj has nothing to gain by deviating from ai

AND

Player II expecting Player I to Play ai has nothing to gain by deviating from playing Aj.

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Principle IIThe players tend to strategy pairs that

are in equilibrium, i.e. stable

An optimal solution is said to be reached if neither player finds it beneficial to change their strategy.